In this chapter, we investigate the relationship between consonance and dissonance, and simple integer ratios of frequencies.
We saw in Sections 3.2 and 3.5 that when a note on a stringed instrument or a wind instrument sounds at a certain pitch, say with frequency ν, sound is essentially periodic with that frequency. The theory of Fourier series shows that such a sound can be decomposed as a sum of sine waves with various phases, at integer multiples of the frequency ν, as in Bernoulli's solution (3.2.7) to the wave equation. The component of the sound with frequency ν is called the fundamental. The component with frequency mν is called the mth harmonic, or the (m – 1)st overtone. So, for example, if m = 3 we obtain the third harmonic, or the second overtone.
Figure 4.1 represents the series of harmonics based on a fundamental at the C below middle C. The seventh harmonic is actually somewhat flatter than the B♭ above the treble clef. In the modern equally tempered scale, even the third and fifth harmonics are very slightly different from the notes G and E shown above – this is more extensively discussed in Chapter 5.
There is another word which we have been using in this context: the mth partial of a sound is the mth frequency component, counted from the bottom. So, for example, on a clarinet, where only the odd harmonics are present, the first partial is the fundamental, or first harmonic, and the second partial is the third harmonic.